Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 246808, 13 pages doi:10.1155/2010/246808 Research Article Strong Convergence Theorem for a New General System of Variational Inequalities in Banach Spaces S Imnang1, and S Suantai2, Department of Mathematics, Faculty of Science, Thaksin University, Phatthalung Campus, Phatthalung 93110, Thailand Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand Correspondence should be addressed to S Suantai, scmti005@chiangmai.ac.th Received 26 July 2010; Revised December 2010; Accepted 30 December 2010 Academic Editor: S Reich Copyright q 2010 S Imnang and S Suantai This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We introduce a new system of general variational inequalities in Banach spaces The equivalence between this system of variational inequalities and fixed point problems concerning the nonexpansive mapping is established By using this equivalent formulation, we introduce an iterative scheme for finding a solution of the system of variational inequalities in Banach spaces Our main result extends a recent result acheived by Yao, Noor, Noor, Liou, and Yaqoob Introduction Let X be a real Banach space, and X ∗ be its dual space Let U {x ∈ X : x 1} denote the unit sphere of X X is said to be uniformly convex if for each ∈ 0, there exists a constant δ > such that for any x, y ∈ U, x−y ≥ implies x y ≤ − δ 1.1 The norm on X is said to be Gˆ teaux differentiable if the limit a lim t→0 x ty − x t 1.2 Fixed Point Theory and Applications exists for each x, y ∈ U and in this case X is said to have a uniformly Frechet differentiable norm if the limit 1.2 is attained uniformly for x, y ∈ U and in this case X is said to be uniformly smooth We define a function ρ : 0, ∞ → 0, ∞ , called the modulus of smoothness of X, as follows: ρ τ sup x y 1, y − : x, y ∈ X, x x−y τ 1.3 It is known that X is uniformly smooth if and only if limτ → ρ τ /τ Let q be a fixed real number with < q ≤ Then a Banach space X is said to be q-uniformly smooth if there exists a constant c > such that ρ τ ≤ cτ q for all τ > For q > 1, the generalized duality mapping ∗ Jq : X → 2X is defined by Jq x f ∈ X ∗ : x, f x q, f x q−1 , ∀x ∈ X 1.4 In particular, if q 2, the mapping J2 is called the normalized duality mapping and usually, we write J2 J If X is a Hilbert space, then J I Further, we have the following properties of the generalized duality mapping Jq : q−2 Jq x x Jq tx t q−1 Jq −x −Jq x for all x ∈ X J2 x for all x ∈ X with x / 0, Jq x for all x ∈ X and t ∈ 0, ∞ , It is known that if X is smooth, then J is single-valued, which is denoted by j Recall that the duality mapping j is said to be weakly sequentially continuous if for each {xn } ⊂ X with xn → x weakly, we have j xn → j x weakly-∗ We know that if X admits a weakly sequentially continuous duality mapping, then X is smooth For the details, see the work of Gossez and Lami Dozo in Let C be a nonempty closed convex subset of a smooth Banach space X Recall that a mapping A : C → X is said to be accretive if Ax − Ay, j x − y ≥0 1.5 for all x, y ∈ C A mapping A : C → X is said to be α-strongly accretive if there exists a constant α > such that Ax − Ay, j x − y ≥α x−y 1.6 for all x, y ∈ C A mapping A : C → X is said to be α-inverse strongly accretive if there exists a constant α > such that Ax − Ay, j x − y ≥ α Ax − Ay 1.7 for all x, y ∈ C A mapping T : C → C is said to be nonexpansive if Tx − Ty ≤ x − y for all x, y ∈ C The fixed point set of T is denoted by F T : {x ∈ C : Tx x} Fixed Point Theory and Applications Let D be a nonempty subset of C A mapping Q : C → D is said to be sunny if Q Qx t x − Qx Qx, 1.8 whenever Qx t x − Qx ∈ C for x ∈ C and t ≥ A mapping Q : C → D is called a retraction if Qx x for all x ∈ D Furthermore, Q is a sunny nonexpansive retraction from C onto D if Q is a retraction from C onto D which is also sunny and nonexpansive A subset D of C is called a sunny nonexpansive retraction of C if there exists a sunny nonexpansive retraction from C onto D It is well known that if X is a Hilbert space, then a sunny nonexpansive retraction QC is coincident with the metric projection from X onto C Conveying an idea of the classical variational inequality, denoted by VI C, A , is to find an x∗ ∈ C such that Ax∗ , y − x∗ ≥ 0, ∀y ∈ C, 1.9 where X H is a Hilbert space and A is a mapping from C into H The variational inequality has been widely studied in the literature; see, for example, the work of Chang et al in , Zhao and He , Plubtieng and Punpaeng , Yao et al and the references therein Let A, B : C → H be two mappings In 2008, Ceng et al considered the following problem of finding x∗ , y∗ ∈ C × C such that λAy∗ x∗ − y∗ , x − x∗ ≥ 0, ∀x ∈ C, μBx∗ y∗ − x∗ , x − y∗ ≥ 0, ∀x ∈ C, 1.10 which is called a general system of variational inequalities, where λ > and μ > are two constants In particular, if A B, then problem 1.10 reduces to finding x∗ , y∗ ∈ C × C such that λAy∗ x∗ − y∗ , x − x∗ ≥ 0, ∀x ∈ C, μAx∗ y∗ − x∗ , x − y∗ ≥ 0, ∀x ∈ C, 1.11 which is defined by Verma and is called the new system of variational inequalities Further, if we add up the requirement that x∗ y∗ , then problem 1.11 reduces to the classical variational inequality VI C, A In 2006, Aoyama et al first considered the following generalized variational inequality problem in Banach spaces Let A : C → X be an accretive operator Find a point x∗ ∈ C such that Ax∗ , j x − x∗ ≥ 0, ∀x ∈ C 1.12 The problem 1.12 is very interesting as it is connected with the fixed point problem for nonlinear mapping and the problem of finding a zero point of an accretive operator in Banach spaces, see 9–11 and the references therein 4 Fixed Point Theory and Applications Aoyama et al introduced the following iterative algorithm in Banach spaces: x0 yn xn QC xn − λn A xn , an xn x ∈ C, − an yn , 1.13 n ≥ 0, where QC is a sunny nonexpansive retraction from X onto C Then they proved a weak convergence theorem which is generalized simultaneously theorems of Browder and Petryshyn 12 and Gol’shte˘n and Tret’yakov 13 In 2008, Hao 14 obtained a strong ı convergence theorem by using the following iterative algorithm: x0 ∈ C, yn xn bn xn an u − bn QC I − λn Axn , − an yn , 1.14 n ≥ 0, where {an }, {bn } are two sequences in 0, and u ∈ C Very recently, in 2009, Yao et al introduced the following system of general variational inequalities in Banach spaces For given two operators A, B : C → X, they considered the problem of finding x∗ , y∗ ∈ C × C such that Ay∗ x∗ − y ∗ , j x − x∗ ≥ 0, ∀x ∈ C, Bx∗ y ∗ − x∗ , j x − y ∗ ≥ 0, ∀x ∈ C, 1.15 which is called the system of general variational inequalities in a real Banach space They proved a strong convergence theorem by using the following iterative algorithm: x0 ∈ C, yn xn an u bn xn QC xn − Bxn , 1.16 cn QC yn − Ayn , n ≥ 0, where {an }, {bn }, and {cn } are three sequences in 0, and u ∈ C In this paper, motivated and inspired by the idea of Yao et al and Cheng et al First, we introduce the following system of variational inequalities in Banach spaces Let C be a nonempty closed convex subset of a real Banach space X Let Ai : C → X for all i 1, 2, be three mappings We consider the following problem of finding x∗ , y∗ , z∗ ∈ C × C × C such that λ A1 y ∗ x∗ − y ∗ , j x − x∗ ≥ 0, ∀x ∈ C, λ2 A2 z∗ y∗ − z∗ , j x − y∗ ≥ 0, ∀x ∈ C, λ A3 x ∗ z∗ − x∗ , j x − z∗ ≥ 0, ∀x ∈ C, 1.17 Fixed Point Theory and Applications which is called a new general system of variational inequalities in Banach spaces, where λi > for all i 1, 2, In particular, if A3 0, z∗ x∗ , and λi for i 1, 2, 3, then problem 1.17 0, z∗ x∗ , then problem 1.17 reduces to the reduces to problem 1.15 Further, if A3 problem 1.10 in a real Hilbert space Second, we introduce iteration process for finding a solution of a new general system of variational inequalities in a real Banach space Starting with arbitrary points v, x1 ∈ C and let {xn }, {yn }, and {zn } be the sequences generated by zn yn xn an v QC xn − λ3 A3 xn , QC zn − λ2 A2 zn , 1.18 − an − bn QC yn − λ1 A1 yn , bn xn n ≥ 1, where λi > for all i 1, 2, and {an }, {bn } are two sequences in 0, Using the demiclosedness principle for nonexpansive mapping, we will show that the sequence {xn } converges strongly to a solution of a new general system of variational inequalities in Banach spaces under some control conditions Preliminaries In this section, we recall the well known results and give some useful lemmas that will be used in the next section Lemma 2.1 see 15 Let X be a q-uniformly smooth Banach space with ≤ q ≤ Then x y q ≤ x q q y, Jq x Ky q 2.1 for all x, y ∈ X, where K is the q-uniformly smooth constant of X The following lemma concerns the sunny nonexpansive retraction Lemma 2.2 see 16, 17 Let C be a closed convex subset of a smooth Banach space X Let D be a nonempty subset of C and Q : C → D be a retraction Then Q is sunny and nonexpansive if and only if u − Qu, j y − Qu ≤ 0, 2.2 for all u ∈ C and y ∈ D The first result regarding the existence of sunny nonexpansive retractions on the fixed point set of a nonexpansive mapping is due to Bruck 18 Remark 2.3 If X is strictly convex and uniformly smooth and if T : C → C is a nonexpansive mapping having a nonempty fixed point set F T , then there exists a sunny nonexpansive retraction of C onto F T Fixed Point Theory and Applications Lemma 2.4 see 19 Assume {an } is a sequence of nonnegative real numbers such that an ≤ − γn an δn , n ≥ 1, 2.3 where {γn } is a sequence in 0, and {δn } is a sequence such that i ∞ n γn ∞; ii lim supn → ∞ δn /γn ≤ or Then limn → ∞ an ∞ n |δn | < ∞ Lemma 2.5 see 20 Let {xn } and {yn } be bounded sequences in a Banach space X and let {bn } be a sequence in 0, with < lim infn → ∞ bn ≤ lim supn → ∞ bn < Suppose xn 1 − bn yn bn xn for all integers n ≥ and lim supn → ∞ yn − yn − xn − xn ≤ Then, limn → ∞ yn − xn Lemma 2.6 see 21 Let X be a uniformly convex Banach space, C a nonempty closed convex subset of X, and T : C → C be an nonexpansive mapping Then I − T is demiclosed at 0, that is, if xn → x weakly and xn − Txn → strongly, then x ∈ F T Main Results In this section, we establish the equivalence between the new general system of variational inequalities 1.17 and some fixed point problem involving a nonexpansive mapping Using the demiclosedness principle for nonexpansive mapping, we prove that the iterative scheme 1.18 converges strongly to a solution of a new general system of variational inequalities 1.17 in a Banach space under some control conditions In order to prove our main result, the following lemmas are needed The next lemmas are crucial for proving the main theorem Lemma 3.1 Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X Let the mapping A : C → X be α-inverse strongly accretive Then, we have I − λA x − I − λA y ≤ x−y 2λ λK − α Ax − Ay , 3.1 where K is the 2-uniformly smooth constant of X In particular, if α ≥ λK , then I − λA is a nonexpansive mapping Proof Indeed, for all x, y ∈ C, from Lemma 2.1, we have I − λA x − I − λA y x − y − λ Ax − Ay ≤ x−y 2 − 2λ Ax − Ay , j x − y 2K λ2 Ax − Ay ≤ x−y − 2λα Ax − Ay x−y 2λ λK − α 3.2 2K λ Ax − Ay 2 Ax − Ay It is clear that, if α ≥ λK , then I − λA is a nonexpansive mapping Fixed Point Theory and Applications Lemma 3.2 Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X Let QC be the sunny nonexpansive retraction from X onto C Let Ai : C → X be an αi -inverse strongly accretive mapping for i 1, 2, Let G : C → C be a mapping defined by Gx QC QC QC x − λ3 A3 x − λ2 A2 QC x − λ3 A3 x −λ1 A1 QC QC x − λ3 A3 x − λ2 A2 QC x − λ3 A3 x If αi ≥ λi K for all i , ∀x ∈ C 3.3 1, 2, 3, then G : C → C is nonexpansive Proof For all x, y ∈ C, we have G x −G y QC QC QC I − λ3 A3 x − λ2 A2 QC I − λ3 A3 x −λ1 A1 QC QC I − λ3 A3 x − λ2 A2 QC I − λ3 A3 x − QC QC QC I − λ3 A3 y − λ2 A2 QC I − λ3 A3 y − λ1 A1 QC QC I − λ3 A3 y − λ2 A2 QC I − λ3 A3 y ≤ QC QC I − λ3 A3 x − λ2 A2 QC I − λ3 A3 x − λ1 A1 QC QC I − λ3 A3 x − λ2 A2 QC I − λ3 A3 x 3.4 − QC QC I − λ3 A3 y − λ2 A2 QC I − λ3 A3 y − λ1 A1 QC QC I − λ3 A3 y − λ2 A2 QC I − λ3 A3 y I − λ1 A1 QC I − λ2 A2 QC I − λ3 A3 x − I − λ1 A1 QC I − λ2 A2 QC I − λ3 A3 y From Lemma 3.1, we have I −λ1 A1 QC I −λ2 A2 QC I −λ3 A3 is nonexpansive which implies by 3.4 that G is nonexpansive Lemma 3.3 Let C be a nonempty closed convex subset of a real smooth Banach space X Let QC be the sunny nonexpansive retraction from X onto C Let Ai : C → X be three nonlinear mappings For given x∗ , y∗ , z∗ ∈ C × C × C, x∗ , y∗ , z∗ is a solution of problem 1.17 if and only if x∗ ∈ F G , y∗ QC z∗ − λ2 A2 z∗ and z∗ QC x∗ − λ3 A3 x∗ , where G is the mapping defined as in Lemma 3.2 Proof Note that we can rewrite 1.17 as x ∗ − y ∗ − λ A1 y ∗ , j t − x ∗ ≥ 0, ∀t ∈ C, y∗ − z∗ − λ2 A2 z∗ , j t − y∗ ≥ 0, ∀t ∈ C, z∗ − x∗ − λ3 A3 x∗ , j t − z∗ ≥ 0, ∀t ∈ C 3.5 Fixed Point Theory and Applications From Lemma 2.2, we can deduce that 3.5 is equivalent to x∗ QC y∗ − λ1 A1 y∗ , y∗ QC z∗ − λ2 A2 z∗ , z∗ QC x∗ − λ3 A3 x∗ It is easy to see that 3.6 is equivalent to x∗ λ A3 x ∗ Gx∗ , y∗ 3.6 QC z∗ − λ2 A2 z∗ and z∗ QC x∗ − From now on we denote by Ω∗ the set of all fixed points of the mapping G Now we prove the strong convergence theorem of algorithm 1.18 for solving problem 1.17 Theorem 3.4 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space X which admits a weakly sequentially continuous duality mapping Let QC be the sunny nonexpansive retraction from X onto C Let the mappings Ai : C → X be αi -inverse strongly accretive with αi ≥ λi K , for all i 1, 2, and Ω∗ / ∅ For given x1 , v ∈ C, let the sequence {xn } be generated iteratively by 1.18 Suppose the sequences {an } and {bn } are two sequences in 0, such that C1 limn → ∞ an and ∞ n an ∞; C2 < lim infn → ∞ bn ≤ lim supn → ∞ bn < Then {xn } converges strongly to Q v where Q is the sunny nonexpansive retraction of C onto Ω∗ Proof Let x∗ ∈ Ω∗ and tn x∗ QC yn − λ1 A1 yn , it follows from Lemma 3.3 that QC QC QC x∗ − λ3 A3 x∗ − λ2 A2 QC x∗ − λ3 A3 x∗ −λ1 A1 QC QC x∗ − λ3 A3 x∗ − λ2 A2 QC x∗ − λ3 A3 x∗ Put y∗ QC z∗ − λ2 A2 z∗ and z∗ xn QC x∗ − λ3 A3 x∗ Then x∗ an v From Lemma 3.1, we have I − λi Ai i tn − x ∗ bn xn 3.7 QC y∗ − λ1 A1 y∗ and − an − bn tn 3.8 1, 2, is nonexpansive Therefore QC yn − λ1 A1 yn − QC y∗ − λ1 A1 y∗ ≤ yn − y∗ QC zn − λ2 A2 zn − QC z∗ − λ2 A2 z∗ ≤ zn − z∗ QC xn − λ3 A3 xn − QC x∗ − λ3 A3 x∗ ≤ xn − x∗ 3.9 Fixed Point Theory and Applications It follows that xn − x∗ an v − an − bn tn − x∗ bn xn ≤ an v − x∗ bn xn − x∗ − an − bn tn − x∗ ≤ an v − x∗ bn xn − x∗ − an − bn xn − x∗ an v − x∗ 3.10 − an xn − x∗ By induction, we have xn − x∗ ≤ max{ v − x∗ , x1 − x∗ } 3.11 Therefore, {xn } is bounded Hence {yn }, {zn }, {tn }, {A1 yn }, {A2 zn }, and {A3 xn } are also bounded By nonexpansiveness of QC and I − λi Ai i 1, 2, , we have tn QC yn − tn ≤ yn 1 Let wn xn 1 − λ2 A2 zn 1 − wn xn an v an 1 − bn − xn v − tn 1 − bn wn for all n ∈ Ỉ and bn xn 1 − bn xn − bn − an − bn − bn 1 − QC xn − λ3 A3 xn 3.12 − xn − bn xn − bn − QC zn − λ2 A2 zn − λ3 A3 xn − bn xn / − bn , n ∈ Ỉ Then xn wn − zn QC xn ≤ xn − QC yn − λ1 A1 yn − yn QC zn ≤ zn − λ1 A1 yn tn 1 − an v − an − bn tn − bn an tn − v − bn tn 3.13 − tn By 3.12 and 3.13 , we have wn − wn − xn − xn ≤ an 1 − bn tn ≤ an 1 − bn v − tn − tn − xn v − tn an tn − v − bn 1 − xn an tn − v − bn 3.14 10 Fixed Point Theory and Applications This together with C1 and C2 , we obtain that lim sup wn n→∞ − wn − xn − xn ≤ 3.15 Hence, by Lemma 2.5, we get xn − wn → as n → ∞ Consequently, lim xn n→∞ − xn lim − bn wn − xn 3.16 − an − bn tn − xn , 3.17 n→∞ Since xn − xn an v − xn therefore tn − xn −→ as n −→ ∞ 3.18 Furthermore, by Lemma 3.2, we have G : C → C is nonexpansive Thus, we have tn − G tn QC yn − λ1 A1 yn − G tn QC QC zn − λ2 A2 zn − λ1 A1 QC zn − λ2 A2 zn − G tn QC QC QC xn − λ3 A3 xn − λ2 A2 QC xn − λ3 A3 xn − λ1 A1 QC QC xn − λ3 A3 xn − λ2 A2 QC xn − λ3 A3 xn G xn − G tn − G tn ≤ xn − tn , 3.19 which implies tn − G tn Since → as n → ∞ xn − G xn ≤ xn − tn tn − G tn G tn − G xn ≤ xn − tn tn − G tn tn − xn , 3.20 therefore lim xn − G xn n→∞ 3.21 Let Q be the sunny nonexpansive retraction of C onto Ω∗ Now we show that lim sup v − Q v, j xn − Q v n→∞ ≤ 3.22 Fixed Point Theory and Applications 11 To prove 3.22 , since {xn } is bounded, we can choose a subsequence {xni } of {xn } which converges weakly to x and lim sup v − Q v, j xn − Q v lim v − Q v, j xni − Q v 3.23 i→∞ n→∞ From Lemma 2.6 and 3.21 , we obtain x ∈ Ω∗ Now, from Lemma 2.2, 3.23 , and the weakly sequential continuity of the duality mapping j, we have lim v − Q v, j xni − Q v lim sup v − Q v, j xn − Q v i→∞ n→∞ 3.24 v − Q v, j x − Q v ≤ From 3.9 , we have xn −Qv an v bn xn − an − bn tn − Q v, j xn an v − Q v, j xn −Qv − an − bn an v − Q v, j xn − an − bn ≤ an v − Q v, j xn tn − Q v j xn −Qv bn tn − Q v 1 − an − bn ≤ an v − Q v, j xn −Qv xn − Q v xn −Qv 1 xn − Q v j xn −Qv xn −Qv −Qv bn 2 −Qv xn − Q v xn 1 − an −Qv xn −Qv 2 xn − Q v xn −Qv −Qv bn −Qv tn − Q v 1 − an − bn an v − Q v, j xn xn − Q v bn 1 −Qv −Qv −Qv bn xn − Q v, j xn − an − bn tn − Q v, j xn ≤ an v − Q v, j xn xn −Qv 2 xn − Q v xn −Qv , 3.25 which implies that xn −Qv ≤ − an xn − Q v 2an v − Q v, j xn −Qv 3.26 It follows from Lemma 2.4, 3.24 , and 3.26 that {xn } converges strongly to Q v This completes the proof 12 Fixed Point Theory and Applications and λi Letting A3 for i 1, 2, in Theorem 3.4, we obtain the following result Corollary 3.5 see 5, Theorem 3.1 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space X which admits a weakly sequentially continuous duality mapping Let QC be the sunny nonexpansive retraction from X onto C Let the mappings Ai : C → X be αi -inverse strongly accretive with αi ≥ K , for all i 1, and Ω∗ / ∅ For given x1 , v ∈ C, and let {xn }, {yn } be the sequences generated by yn xn an v bn xn QC xn − A2 xn , − an − bn QC yn − A1 yn , n ≥ 1, 3.27 where {an }, {bn } are two sequences in 0, such that C1 limn → ∞ an and ∞ n an ∞; C2 < lim infn → ∞ bn ≤ lim supn → ∞ bn < Then {xn } converges strongly to Q v where Q is the sunny nonexpansive retraction of C onto Ω∗ Acknowledgments The authors wish to express their gratitude to the referees for careful reading of the manuscript and helpful suggestions The authors would like to thank the Commission on Higher Education, the Thailand Research Fund, the Thaksin university, the Centre of Excellence in Mathematics, and the Graduate School of Chiang Mai University, Thailand for their financial support References J.-P Gossez and E Lami Dozo, “Some geometric properties related to the fixed point theory for 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